Weak breaking of translational symmetry in Z2 Topological ordered states

We study Z2 topologically ordered states enriched by translational symmetry by employing a recently developed 2D bosonization approach that implements an exact Z2 charge-flux attachement in the lattice. With this we develop a general description of a series of anomalous properties of these states, such as ground state degeneracy that depends on system size, the emergence of dangling Majorana modes and the 'weak symmetry breaking’ of translations. We demonstrate that this ‘weak symmetry breaking' of translations appears in certain states that are weak topological superconductors of the epsilon fermionic spinons, where they form stacks of Kitaev wires. This leads to the amusing property that there is no local operator that can translate the flux across a single Kitaev wire of fermonic spinons without paying an energy gap in spite of the vacuum remaining fully translational invariant. By extending the Z2 charge-flux attachment to open lattices and cylinders we construct a plethora of exactly solvable models providing an exact description of their dispersive Majorana gapless boundary modes.